Spherical Black Holes cannot Support Scalar Hair

TL;DR

This paper critically examines the BBMB black hole solution and demonstrates that spherical black holes cannot support classical scalar hair, reinforcing the no-hair conjecture by showing the solution's inconsistencies.

Contribution

It provides a detailed critique of the BBMB solution, showing it is not a valid black hole solution and supports the idea that spherical black holes lack scalar hair.

Findings

The stress energy tensor is ill-defined at the horizon.

Regularization yields ambiguous results for the stress energy tensor.

Spherical black holes cannot carry classical scalar hair.

Abstract

The static spherically symmetric ``black hole solution" of the Einstein - conformally invariant massless scalar field equations known as the BBMB ( Bocharova, , Bronikov, Melinkov, Bekenstein) black hole is critically examined. It is shown that the stress energy tensor is ill-defined at the horizon, and that its evaluation through suitable regularization yields ambiguous results. Consequently, the configuration fails to represent a genuine black hole solution. With the removal of this solution as a counterexample to the no hair conjecture, we argue that the following appears to be true: Spherical black holes cannot carry any kind of classical scalar hair.